Table Of ContentRepetitive Motion Planning and Control
of Redundant Robot Manipulators
Yunong Zhang (cid:2) Zhijun Zhang
Repetitive Motion Planning
and Control of Redundant
Robot Manipulators
YunongZhang
ZhijunZhang
SunYat-senUniversity
Guangzhou
People’sRepublicofChina
ISBN978-3-642-37517-0 ISBN978-3-642-37518-7(eBook)
DOI10.1007/978-3-642-37518-7
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013938762
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Preface
Kinematically redundant manipulators are those having more degrees of freedom
(DOF)thanrequiredtoperformagivenend-effectorprimarytask.Onefundamen-
tal issue in operating such a robot system is the redundancy-resolution problem
(orso-calledinversekinematicsproblem).Aninverse-kinematicsschemeiscalled
repetitive if it maps closed paths in the task space (i.e., cyclic sequences of tasks)
toclosedtrajectoriesintheconfigurationspace(i.e.,cyclicsequencesofconfigura-
tions).Nonrepetitiveproblemisthatthejointanglesmaynotreturntotheirinitial
values after the end-effector traces a closed path in its workspace. Nonrepetitive
problem results in a joint-angle drift phenomenon and may induce a problem that
themanipulator’sbehaviorishardtopredict;itisthenlessefficienttoreadjustthe
manipulator’sconfigurationaftereverycycle.
In this book we present four typical motion planning schemes. One of them is
online repetitive motion planning (RMP) scheme, or so-called cyclic motion gen-
eration(CMG)scheme,anditsproofbasedonneuraldynamicmethodsisgivenin
detail.Then,someotheroptimizationschemes,whichcanbeviewedastheexten-
sionsoftheRMPscheme,aredevelopedandinvestigatedforthepurposeofrepet-
itive motion planning. Furthermore, we unify them as quadratic programs (QPs).
Moreover, some important QP solvers (including neural networks and numerical
algorithms) are employed to solve the resultant QPs. Computer-simulation results
basedonvariousroboticmodelsdemonstratetheeffectivenessoftheproposedRMP
schemes.For substantiatingthe physical realizabilityof the RMP schemes via QP
formulation,oneoftheRMPschemesisappliedtoanactualplanarsix-DOFrobot
manipulator(ortosay,robotarm).Morespecifically,wehavethefollowingorgani-
zation.
InChap.1,wefirstdiscusstheimportanceoftheRMPresearchandreviewthe
recent results on RMP. Then we present three kinds of manipulators. Their Jaco-
bian matrices are derived in detail for further discussion, simulations, and experi-
ments.
InChap.2,anoptimizationschemeisfirstpresentedandinvestigatedforonline
RMPofredundantrobotmanipulators.Then,threeextendedschemesaredeveloped
vii
viii Preface
forthepurposeofRMP.Thesefourschemesarefinallyreformulatedandunifiedas
QPproblemswithdifferentdefinitionsofthesamecoefficients.
In Chap. 3, we employ the well-known gradient-based neural network (GNN)
and Zhang neural network (ZNN) approaches to analyze and prove the repetitive
motionperformanceindex.
Oneofstate-of-the-artrecurrentneuralnetworks(RNN)isdualneuralnetwork
(DNN).ItcansolveQPinrealtime.TheDNNisofsimplepiecewiselineardynam-
ics and has global (exponential) convergence to optimal solutions. In Chap. 4, we
presentsuchaDNNanditsdesignmethod.
InadditiontoDNN,anothertypeofneuralnetworkhasbeenwidelyusedinre-
centyears,i.e.,primal–dualneuralnetwork(PDNN).Thiskindofnetworkincludes
mainlyatraditionalprimal–dualneuralnetwork,alinearvariationalinequality(LVI)
basedPDNN,andasimplifiedLVI-basedPDNN,presentedinChap.5.
Different from the neural-network-based QP solvers, the numerical algorithms
asQPsolversareveryimportantaswell,especiallyinthereal-timekinematicreso-
lutionviatoday’sdigitalcomputers.InChaps.6and7,wepresentnumericalalgo-
rithms94LVIandE47,respectively.
InChaps.8,9,and10,wepresentcomputersimulationexamplesbasedonplanar
multilinkmanipulators,PUMA560,and PA10 robots to demonstrate the effective-
nessofthepresentedRMPschemeandthecorrespondingQPsolvers.
To demonstrate the hardware realizability and efficacy of the QP-based meth-
odsforsolvingthenonrepetitiveproblem,Chap.11givesarepetitivemotionplan-
ning and control (RMPC) scheme, and realizes this scheme on a physical pla-
nar six-DOF push-rod-joint (PRJ) manipulator. To control the real PRJ manipula-
tor,thisschemeconsidersvariablejoint-velocitylimitsandjoint-limitmargins.To
decrease the model disturbance and computational round-off errors, this scheme
also considers the position-error feedback. Then, the scheme is reformulated as
a QP problem. Due to control via a digital computer, a discrete-time QP solver,
termed piecewise linear equation (PLE) based numerical algorithm (i.e., numer-
ical algorithm 94LVI), is presented to solve the QP problem. For comparison,
both of the nonrepetitive and repetitive motions are performed on the six-DOF
PRJ manipulator to track square, B-shaped, and circular paths. Theoretical anal-
ysis and experimental results validate the physical realizability and effectiveness
of the RMPC scheme. Position-error analysis further verifies the accuracy of this
scheme.
In summary, the book solves the nonrepetitive motion problem, which has
stood in the areas of robotics and control for 30 years (specifically, since the
work of Klein and Huang in 1983). The QP technique is exploited in this RMPC
research with rich verification of simulations and experiments, while the tradi-
tional methods relate to pseudoinverse-type solutions. Now, the door to the indus-
trial applications of redundant manipulators is completely open, as the difficult
nonrepetitive motion problem has been solved truly, systematically, and method-
ologically. Without doubt, this book can be extended. Any comments or sugges-
tionsare welcome.The authorscanbe contactedviae-mails [email protected].
Preface ix
[email protected]://sist.
sysu.edu.cn/~zhynong/.
Guangzhou,China YunongZhang
February2013 ZhijunZhang
Acknowledgements
During the work on this book, we have had the pleasure of discussing its various
aspectsandresultswithmanycooperatorsandstudents.Wehighlyappreciatetheir
contributions, which particularly allowed us to improve our book manuscript. Es-
pecially valuable help was provided by Dongsheng Guo, Huarong Wu, Lin Xiao,
BingguoMu,XiaotianYu,JunLi,SenboFu,JinhaoChen,YonghuaYin,andWeib-
ingLi.Wearegratefultothemfortheirhelpandsuggestions.
The continuous support of our research by the National Science Foundation of
China under Grants 61075121 and 60935001 and by the Fundamental Research
FundsfortheCentralUniversitiesofChinaisgratefullyacknowledgedhereeither.
Besides, we would like to thank sincerely the editors for their very important
andconstructivecommentsandsuggestionsprovided,inadditiontotheirtimeand
effortsspentinhandlingthisbook.
Wearealwaysverygratefultothenicepeople(especiallythestaffinSpringer)
for their strong support and encouragement during the preparation and publishing
ofthisbook.
xi
Contents
1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 RedundancyoftheRobots . . . . . . . . . . . . . . . . . . 1
1.1.2 NeuralNetworksforResolvingtheRedundancy . . . . . . 2
1.1.3 NumericalAlgorithmsforResolvingtheRedundancy . . . 2
1.1.4 NonrepetitiveMotionProblem . . . . . . . . . . . . . . . 3
1.2 ForwardKinematicEquation . . . . . . . . . . . . . . . . . . . . 3
1.3 GeometryofFive-LinkPlanarRobotManipulator . . . . . . . . . 4
1.4 GeometryofPA10 . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 GeometryofPUMA560 . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 ChapterConclusions . . . . . . . . . . . . . . . . . . . . . . . . . 12
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
PartI Schemes
2 RoboticRMPSchemesandQPFormulations . . . . . . . . . . . . . 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 PhysicallyConstrainedRMPScheme . . . . . . . . . . . . . . . . 18
2.3 ExtensionsofPhysicallyConstrainedRMPScheme . . . . . . . . 20
2.3.1 ExtensionI . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 ExtensionII . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.3 ExtensionIII . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 QPReformulationandUnification . . . . . . . . . . . . . . . . . 22
2.5 ChapterConclusions . . . . . . . . . . . . . . . . . . . . . . . . . 24
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 ProofsofRepetitiveMotionPerformanceIndex . . . . . . . . . . . . 27
3.1 ProofviaGNNApproach . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 BackgroundofGNN . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 ProofviaZNNApproach . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 BackgroundofZNN . . . . . . . . . . . . . . . . . . . . . 28
xiii
